Flow regimes identification of air water counter current flow in vertical annulus using differential pressure signals and machine learning

This study investigates the gas–liquid two-phase counter-current flow through a vertical annulus, a phenomenon prevalent across numerous industrial fields. The presence of an inner pipe and varying degrees of eccentricity between the inner and outer pipes often blur the clear demarcation of flow regime boundaries. To address this, we designed a vertical annulus with adjustable eccentricity (outer and inner diameters of 125 mm and 75 mm, respectively). We conducted gas–liquid counter-current flow experiments under specific conditions: gas superficial velocity ranging from 0.06 to 5.04 m/s, liquid superficial velocity from 0.01 to 0.25 m/s, and five levels of eccentricity (e = 0, 0.25, 0.5, 0.75, 1). We collected differential pressure data at two distinct height distances (DP1: 50 mm and DP2: 1000 mm). We used vectors, composed of both the probability density functions (PDFs) of the differential pressure signals and the power spectral density (PSD) reduced via Principal Component Analysis, as features. Using the CFDP clustering algorithm—based on local density—we clustered the flow regimes of the experimental data, thereby achieving an objective and consistent identification of the flow regime of gas–liquid two-phase counter-current flow in a vertical annulus. Our analysis reveals that for DP1, the main differences in the PSD of various flow regimes occur within the 0.5–1 Hz range. Among the three flow regimes involved, the slug flow exhibits the highest power intensity, followed by the bubbly flow, with the churn flow having the least. In terms of differential pressure distribution, the bubbly and churn flows have a concentrated distribution, while the slug flow is more dispersed. For DP2, the PSD differences primarily exist within the 0.5–2 Hz range. The churn flow has the highest power intensity, followed by the slug flow, with the bubbly flow being the weakest. Here, the bubbly flow's differential pressure distribution is concentrated, while the slug and churn flows are more dispersed. Based on the results of the flow regime classification, we generated a flow regime map and analyzed the influence of annulus eccentricity on the flow regime. We found that in most cases, pipe eccentricity does not significantly affect the flow regime. However, in the transition region—such as the bubbly to slug flow transition zone—flows with medium eccentricity values (e = 0.5, 0.75) are more likely to transition to slug flow. We compared the visual recognition results of flow regimes with the clustering results. 4.04% of the total samples showed different results from visual recognition and clustering, primarily located in the flow regime transition area. Since visually distinguishing flow regimes in these areas is typically challenging, our methodology offers an objective classification approach for gas–liquid two-phase counter-current flow in a vertical annulus.

Homo,Hete, P-H,F 0.170 0.070 7.0 Counter-current 0 Vertical Air/water direction of the pipe.The movement of bubbles causes fluctuations in the differential pressure between two points along the pipe's axis.Numerous studies have shown that the frequency and time domain features of the differential pressure signal are related to the two-phase flow regime 41,42 .For example, flow regime classification can be achieved based on the PSD of the differential pressure signal 20,38,43 , and the PDF based on differential pressure numerical interval statistics 15,41,44,45 .Simultaneously, the distance of differential pressure signal collection is related to the two-phase flow characteristics it reflects 15,20 .Differential pressure signals collected at smaller distances (such as D/2) are more sensitive to smaller scale two-phase flow characteristics (such as individual bubbles and shorter Taylor bubbles).In contrast, those collected at larger distances are more sensitive to larger scale two-phase flow characteristics (such as pressure fluctuations in agitated flow).This study will use differential pressure signals collected at two different distances to obtain a more comprehensive reflection of two-phase flow characteristics.
Considering the above discussion, due to the complexity of gas-liquid counter-current flow in vertical annuli, its detection signal features have high dimensions.To avoid the subjective factors introduced in the manual flow regime identification process, many unsupervised machine learning classification methods have been applied in the field of gas-liquid two-phase flow regime recognition, such as K-means 34,46 , Fuzzy C -means 47 , selforganizing maps (SOM) 48 , expectation maximization (EM) 49 .These methods do not rely on prior information like flow regime labels, but solely on distance measures between experimental samples to classify flow regimes.The correspondence between classifications and flow regimes is then achieved by comparing the features of each classification.This study will be conducted based on this approach.

Experimental equipment and data acquisition
To study the problem of identifying flow regimes of gas-liquid two-phase counter-current flow in annuli at different eccentricities, a set of experimental pipe columns and auxiliary experimental equipment with continuously adjustable eccentricity was designed.The structure is described as follows: The main body of the experimental pipe column is composed of two layers of acrylic pipes with a height of 6 m.The inner diameter of the outer pipe is 125 mm, and the outer diameter of the inner pipe is 75 mm.The inner pipe is sealed at both ends and only plays a role of occupying space, with the gas and liquid flowing in the annular space between the inner and outer pipes.An air inlet and a liquid outlet are set at the lower end of the outer pipe, and the upper end is connected to a gas-liquid separator to supply water to the pipe column and discharge gas.
The experimental medium is water and air.During the experiment, the pump supplies liquid to the water tank, the water enters the pipe column from the water tank, and then is discharged through the liquid outlet and the flow rate is measured.The air enters the experimental pipe column from the bottom of the pipe column through the gas sparger after being measured in the pipeline from the compressor.After gas-liquid separation in the top water tank, it enters the atmosphere.Sensors for pressure, differential pressure, and temperature are installed on the pipe column.The overall structure of the system and the installation positions of the sensors are shown in Fig. 1.
The inner and outer pipes of the experimental pipe column are connected by two sets of T-shaped lead screws and slide sleeves at the 2.0 m and 4.0 m heights (from the bottom of the outer pipe).The slide sleeve is perpendicular to the pipeline axis and fixed on the inner pipe.The T-shaped lead screw pass through the slide sleeve and both ends pass through a flange ring clamped between the outer pipe flange joints, achieving a rotating seal.Therefore, by rotating the lead screws, the relative positions of the inner and outer pipes can be changed.
The exposed ends of the lead screws are each fitted with a synchronous pulley.The synchronous pulley are connected by a synchronous belt, which can achieve synchronized rotation.Therefore, by pulling the synchronous belt, the inner pipe can be moved parallel to the outer pipe, achieving a continuous adjustment of the pipe column eccentricity within the range of 0-1, as shown in Fig. 2 The eccentricity is defined 10 as the ratio of the distance between the axes of the inner and outer pipes to the difference in their diameter, as expressed in Eq. ( 1) During the gas-liquid two phase counter-current flow, liquid flooding may occur when the liquid phase velocity is large, Sutton et al. 50conducted experimental studies on the entrainment phenomenon of downward flowing liquid on gas.Through experiments with air and water, they provided a boundary for liquid flooding, which occurs when the superficial velocity of the liquid phase exceeds 0.152 m/s.Further experimental research by Wu et al. 23 pointed out that the flooding phenomenon of the gas and liquid phase may occur simultaneously in local areas, and the boundary for liquid flooding may be lower.
To prevent gas from entering the liquid discharge pipe and affecting the measurement of the liquid phase, a cylindrical gas separator densely covered with small holes was designed at the inlet of the liquid discharge pipe to prevent bubbles from entering.Through theoretical calculations and experimental verification, it can guarantee that no large bubbles will enter the liquid discharge pipeline when the superficial velocity of the liquid phase does not exceed 0.28 m/sec.
Experiments were conducted on the above experimental system under different combinations of gas, liquid flow rates, and eccentricities.All experiments were conducted under room temperature conditions (22 °C).Multiple flow combinations were selected within the range of gas superficial velocity Vsg = 0.06-5.04m/s (standard conditions) and liquid superficial velocity Vsl = 0.01-0.25 m/s.Each flow combination was tested five times under e = 0, 0.25, 0.5, 0.75, 1.0 conditions, and eventually 495 groups of usable experimental data were obtained after data cleaning.The Dp1 signal is collected by a differential pressure sensor with a measurement range of − 2.5-+ 2.5KP and an error of 0.1%.The Dp2 signal is obtained by subtracting the data synchronously collected by pressure sensors Pre1 and Pre2, both of which have a range of 20kPa and an error of 0.5%.All pressure taps are vertically arranged on the outer pipe wall, with their projection points coinciding on the annular cross-section.The distance between the Dp1 pressure taps is 50 mm, and the distance between the Pre1 and Pre2 pressure taps is 1000 mm, as shown in Fig. 1.
The liquid phase flow meter has a range of 1-10 m 3 /h with an error of 0.5%.The gas phase flow meter I has a range of 1-12 m 3 /h with an error of 1%, and the gas phase flow meter II has a range of 10-150 m 3 /h with an error of 1.5%.
The sampling rate of the sensor signal was 100sps with a sampling duration of 75s.All signals were transmitted using a 4-20 mA method.The transmission cable used a 0.5 mm 2 shielded twisted pair, the sampling resistance was 150Ω with an accuracy of 0.01%, and the data acquisition card was a National Instruments USB-6361.All frequency converter input and output ends were equipped with filtering devices to eliminate electromagnetic interference from the frequency converter.The work of several researchers has shown that there is a correlation between the features of the PSD of differential pressure fluctuations 15,44 and the state of gas-liquid two-phase flow.However, the fluctuation features of different flow regimes differ only in certain frequency bands.Drahos̆ and Cȇrmák 51 as well as Letzel et al. 52 pointed out that in circular pipes and bubble columns, the contribution of pressure fluctuations caused by bubble movement to the differential pressure spectrum is mainly concentrated in the 1-10 Hz range.Jaiboon et al. 43 pointed out that the "slug" structure forms a peak in the 1.5-2 Hz area on the differential pressure signal spectrum.Benjamin Wu et al. 23 found that individual small bubbles appearing from the air sparger are related to the peak near 11Hz on the differential pressure spectrum, and the flow of liquid also has an impact on this peak.
To highlight the main differences in pressure fluctuations under different flow regimes on the spectrum, the PCA technique was used to perform dimensionality reduction on the PSD.Before using the PCA technique, it is necessary to determine the Explained Variance Ratio (EVR) to be retained.Its value determines the data dimension after dimension reduction.The number of statistical intervals bins of the PDF also needs to be determined.During the research process, with the dataset shuffle seed and other algorithm parameters fixed, experiments were conducted at several levels of EVR at 0.8, 0.85, 0.9, 0.95 and bins at 50, 75, 100, 125, 150.The method is to gradually increase the EVR or bins until the average distance of the cluster centers no longer increases rapidly, that is, the value at the inflection point of the gain increase rate is chosen.
The final determined EVR value is 0.9, which reduces the dimensions of the PSD of the DP1 and DP2 signals from 376 dimensions (corresponding to a sampling duration of 75 s and a sampling rate of 100 sps) to 79 and 87 dimensions, as shown in Fig. 3.
The number of statistical intervals for the PDF is determined to be bins = 100.Therefore, the dimensionreduced PSD (79 dimensions + 87 dimensions) and PDF (100 dimensions + 100 dimensions), a total of 366 dimensions, are used as the distance measure between test samples.After normalization, this forms the dataset for subsequent machine learning.The overall process is shown in Fig. 4.

Clustering
To avoid the subjectivity of flow regime classification through visual imaging, this paper uses an unsupervised machine learning method, namely clustering, to cluster the flow regimes involved in annular gas-liquid twophase flow, in order to obtain a consistent and objective definition and description of flow regime characteristics.
In the past several decades, numerous clustering methods have been developed by different scholars 53 .As discussed in the first part, some classical methods have been applied to multiphase flow regime recognition problems.However, these clustering methods have two limitations: firstly, these classification algorithms make preliminary assumptions about the distribution structure of samples in the feature space.For example, the k-means algorithm assumes that the distribution of the same category in the feature space is spherical, which may not necessarily be consistent with reality.Secondly, these classification algorithms require manual specification of some hyper-parameter values, the selection of which has a significant impact on the classification results.For  www.nature.com/scientificreports/example, the number of classifications corresponding to the number of flow regimes, the artificially specified results are difficult to reflect the inherent distribution characteristics between categories.
To avoid the shortcomings of the above clustering methods, in this study, a clustering method based on p-density peaks, "Clustering by Fast Search and Find of Density Peaks (CFDP)", was used.It was proposed by Alex Rodriguez and Alessandro Laio 54 .Its basic assumption is that the class center is surrounded by neighbors with lower local density and is far away from those nodes with higher local density.The local density and distance are defined as follows: The distance δ i from the class center to a node with higher local density is defined as: For the point with the highest local density, take: Determination of the cluster center.One of the advantages of this algorithm is the determination of the cluster center.After drawing the design map, the number of cluster centers with larger ρ and δ values will be intuitively displayed.
In the case of a smaller dataset, the above algorithm can use the Gaussian kernel function to replace the truncated kernel function in Eqs. ( 2) and (3) to enhance the robustness of the algorithm.This paper uses formula (6)  to calculate the local density, defined as follows:

Clustering results analysis
Following the steps of the CFDP algorithm in "Feature extraction" section, all test samples are plotted in the ρ − δ coordinate system to obtain the Design Map as shown in Fig. 5.Among them, the ρ * δ values of three points are significantly larger than other points, showing that there are three points with a clear advantage to become the cluster center (red points).This is the same as the number of three types of flow regimes involved in this paper.After determining them as the cluster centers, the category of the remaining points (blue) is determined in turn according to the principle in "Feature extraction" section.Finally, the number of test points corresponding to the three classifications is determined to be 120, 297, and 78 respectively.
Figures 6 and 7 show the distribution of the clustering results on the first three main components in the space after the PSD of the DP1 and DP2 signals is reduced.It can be seen that, firstly, whether it is on the DP1 (2) signal or on the DP2 signal, the three categories have not been completely separated.This indicates that the three types of flow regimes cannot be distinguished using only PSD features, which verifies the necessity of adding the PDF to the features.Secondly, the distribution of clusters is obviously non-convex.This distribution makes those clustering algorithms that assume the high-dimensional spherical shape of the same cluster distribution no longer applicable.However, it can be observed that the same cluster has continuity in distribution density.This confirms the necessity of using the CFDP algorithm.

Flow regimes identification results
The three categories correspond to three flow regimes: bubbly flow, slug flow, and churn flow, respectively.Figures 8 and 9 are the average PSD and PDF plots of the two differential pressure signals corresponding to the three flow regimes.As shown in Fig. 8, the main differences in the PSD curves corresponding to the three different flow regimes are all distributed below 5 Hz.According to the conclusions of Drahoš et al. 55 , the part below 0.5 Hz on the PSD curve is caused by environmental factors such as mechanical vibrations, and the differences in this part can be ignored.For the DP1 signal, as shown in Fig. 8a, the differences between the three different flow regimes are mainly in the 0.5-1 Hz range.In this frequency band, slug flow has the maximum power, followed by bubbly flow, and churn flow has the least power.The solid circles are the positions of the maximum values of the three curves.It can be seen that the frequencies corresponding to the maximum power of the three flow regimes are 0.8 Hz, 0.67 Hz, and 0.67 Hz, respectively.
Figure 8b presents the average of the PSD of the DP2 signals corresponding to the three different flow regimes.It can be seen that the differences between different flow regimes primarily lie in the 0.5-2 Hz range.The churn flow possesses the maximum power, followed by slug flow, and bubbly flow has the least power.The frequencies corresponding to the maximum power of the three flow regimes are 0.53 Hz, 0.8 Hz, and 1.2 Hz, respectively.
Comparing Figures (a) and (b), it can be seen that the ranking of power corresponding to three different flow regimes on two different signals is inconsistent, and the frequencies corresponding to the maximum power are not identical.The possible reason for this discrepancy is that the distance for collecting differential pressure signals by DP1 is relatively small (50 mm), which is far less than the scale of the structure of slug flow or churn flow, hence it cannot fully reflect the fluctuation characteristics of these two flow regimes.In Fig. 9b, the distribution curve forms corresponding to slug flow and churn flow are relatively similar, with a larger variance in differential pressure values, and the means exhibit significant differences.In contrast, the differential pressure values corresponding to bubbly flow are more concentrated.In the figure, the vertical black dashed lines represent the range of static water pressure corresponding to the two differential pressure signals.For DP1, this range is 0-0.5 kPa, and for DP2, the range is 0-9.8 kPa.
Comparing the two differential pressure signals, it can be observed that the areas where the DP1 signal exceeds the static pressure range are significantly more than those for DP2.This reveals that the axial differential pressure of two-phase flow in the vertical annulus has greater fluctuations at a smaller scale.

Eccentricity effect analysis
Figure 10 depicts the distribution map of the flow regimes for all test points, where different flow regimes are marked with different colors, and different symbols represent the eccentricity of the pipe column during the test.Here, Vsg represents the gas superficial velocity, and Vsl denotes the liquid superficial velocity.
As can be observed, in most cases, when the superficial velocities of gas and liquid are the same, the same flow regime is present under different eccentricities.However, in some flow regime transition areas, when the superficial velocity of gas and liquid are very close, different flow regimes are classified under different eccentricities.As shown in the subfigure in Fig. 10, it represents test data from five different eccentricities with similar gas-liquid flow rates.At an eccentricity of e = 0.5, although the corresponding gas superficial velocity is less than the cases of e = 0, 0.25, 0.75, and 1, its flow regime is still classified as slug flow instead of bubbly flow.In fact, in the transition area from bubbly flow to slug flow, multiple sets of test data with similar gas-liquid flow rates show that the flow regime is bubbly flow when e = 0, and it transitions to slug flow at other eccentricities.
According to the analysis results in Figs. 8 and 9, under the slug flow state, the average power of low-frequency fluctuations is higher than that of bubbly flow, and the average pressure gradient is lower than that of bubbly flow.However, the differences in the gas flow rates at each test point in the subfigure of Fig. 10 are not significant, hence the pressure gradients are similar.Therefore, the main difference lies in the power of differential pressure fluctuations.
To analyze the differences in differential pressure fluctuations at different eccentricities, a statistical comparison of the variance data of the two differential pressure signals at different eccentricities was performed, as shown in Fig. 11.For both DP1 and DP2 signals, the mean variances corresponding to different eccentricities are smallest when e = 0 and largest when e = 0.75.The range of variance distribution is largest when e = 0.5 and smallest when e = 1.Considering that the larger the variance of the differential pressure values, the stronger the fluctuation, the explanation for the phenomenon displayed in the bubbly flow-slug flow transition area in Fig. 10 is as follows: Although the gas-liquid flow rates are similar, the variance of the differential pressure values is larger when the eccentricity is at a medium value, i.e., e = 0.5, 0.75.Therefore, it is more similar to slug flow, www.nature.com/scientificreports/which has a larger fluctuation power.When e = 0, the variance of differential pressure is the smallest, and it finally transitions to the slug flow stage.This manifestation is consistent with the conclusions of Kelessidis et al. 19 who compared the flow regime transitions in two vertical annuli, concentric (e = 0) and semi-concentric (e = 0.5), and believed that the change in eccentricity does not greatly influence the flow regime transition.However, in the case of large eccentricity rather than small, it was observed that local flow regimes are more likely to develop at lower gas velocities.

Comparison of flow regime clustering results and visual recognition results
Finally, as a test of the accuracy of flow regime recognition by this method, a comparison was made between the flow regime clustering results and visual recognition results.The confusion matrix is shown in Fig. 12.It can be seen that the cases with a large difference between the two methods are mainly: (a) The clustering method  To analyze the reasons for the differences in the results of the two flow regime recognition.methods, samples with different recognition results were plotted on the flow regime map.At the same time, support vector classifiers (SVC) were trained using the superficial velocities and flow regimes of the experimental data corresponding to different eccentricities to divide the areas of different flow regimes.This is shown in Fig. 13, where the green line bundles represent the transition boundary between bubbly flow and slug flow, and the purple line bundles represent the transition boundary between slug flow and churn flow.Different alphas represent different eccentricities.As can be seen, the samples with differences in recognition results are mainly located in the flow regime transition area.Subfigure (a) shows a high-speed photographic image of a sample that is visually Figure 13.Samples with divergent flow regime identification results: (a) Visually recognized as "slug" but identified as "bubbly" by the clustering method (b) Visually recognized as "slug" but identified as "churn" by the clustering method.recognized as "slug" but recognized as "bubbly" by the clustering method.Subfigure (b) shows a sample that is visually recognized as "slug" but recognized as "churn" by the clustering method.
In fact, in vertical annular gas-liquid two-phase flow with larger geometric dimensions, it is difficult for regular Taylor bubbles to appear.Therefore, flow regime classification based on visual judgment is actually not reliable, especially in the flow regime transition area.The accuracy of recognition largely depends on subjective experience and the quality of high-speed photographic images.In this case, the clustering method based on quantitative calculation should have better accuracy and objectivity.

Conclusion
A vertical annular pipe column with adjustable eccentricity between the inner and outer pipes was designed.Gas-liquid two-phase counter-current flow experiments were conducted under different combinations of gas and liquid flow rates and eccentricities, and differential pressure data were collected at two different distances.
The PDF of the two differential pressure signals and the PSD reduced by the PCA method formed the features.The flow regimes of each experimental data were clustered using the density-based clustering algorithm CFDP, achieving an objective and consistent flow regime classification for vertical annular gas-liquid counter-current flow.
The analysis of the two differential pressure signals indicates that for the differential pressure signal DP1 collected at a smaller distance (50 mm), the differences in the PSD of different flow regimes mainly lie in the 0.5-1 Hz range.The power intensities of the three involved flow regimes are strongest in slug flow, followed by bubbly flow, and weakest in churn flow.In terms of differential pressure distribution, the differential pressure distributions of bubbly flow and churn flow are concentrated, while slug flow distribution is more dispersed.For the differential pressure signal DP2 collected at a larger distance (1000 mm), the differences in the PSD of different flow regimes mainly lie in the 0.5-2 Hz range.The power intensities of the three involved flow regimes are strongest in churn flow, followed by slug flow, and weakest in bubbly flow.In terms of differential pressure distribution, the differential pressure distribution of bubbly flow is concentrated, while the distributions of slug flow and churn flow are more dispersed.
Based on the flow regime classification results, a flow regime map was drawn, and the influence of pipe column eccentricity on the flow regime was analyzed.The results show that in most cases, the pipe eccentricity has no significant effect on the flow regime.However, in the flow regime transition area, such as the bubbly flow-slug flow transition area, when the superficial velocities of gas and liquid are not much different, some flow processes with medium eccentricity values, i.e., e = 0.5, 0.75, tend to first transition to slug flow.
A comparison was made between the visual recognition results and clustering results of the flow regime.Samples with different recognition results account for 4.04% of the total and are mainly located in the flow regime transition area, where it is usually difficult to visually identify the flow regime.Therefore, this method provides an objective classification method for annular gas-liquid two-phase flow. https://doi.org/10.1038/s41598-024-63270-x

Figure 1 .
Figure 1.(a) Schematic diagram of the experimental apparatus and (b) Photograph of the experimental apparatus.

Figure 4 .
Figure 4.The flow regime classification process based on dual differential pressure signals and unsupervised machine learning methods.

Figure 5 .
Figure 5. Design Map of the CFDP algorithm.

Figure 6 .Figure 7 .
Figure 6.Distribution of clustering results on the first three principal components of the dimensionalityreduced space of the PSD of the DP1 signal.

Figure 8 .
Figure 8. Distribution of mean PSD values of two differential pressure signals corresponding to three flow regimes.

Figure 9
Figure 9 displays the average PDF of the two differential pressure signals corresponding to the three different flow regimes.As can be seen in Fig. 9a, on the PDF of the DP1 signal, the curve shapes corresponding to bubbly flow and churn flow are similar, with a more concentrated distribution of differential pressure values.The primary difference lies in the mean differential pressure, whereas the curve corresponding to slug flow appears relatively flat and dispersed, indicating a larger variance in differential pressure values and suggesting strong fluctuations during the slug flow stage.In Fig.9b, the distribution curve forms corresponding to slug flow and churn flow are relatively similar, with a larger variance in differential pressure values, and the means exhibit significant differences.In contrast, the differential pressure values corresponding to bubbly flow are more concentrated.In the figure, the vertical black dashed lines represent the range of static water pressure corresponding to the two differential pressure signals.For DP1, this range is 0-0.5 kPa, and for DP2, the range is 0-9.8 kPa.Comparing the two differential pressure signals, it can be observed that the areas where the DP1 signal exceeds the static pressure range are significantly more than those for DP2.This reveals that the axial differential pressure of two-phase flow in the vertical annulus has greater fluctuations at a smaller scale.

Figure 9 .
Figure 9. Distribution of PDF of two differential pressure signals corresponding to three flow regimes.

Figure 10 .
Figure 10.Flow regime map of 125 mm/75 mm annular gas-liquid counter-current flow.In the transition area between bubbly flow and slug flow, in some pipe columns with intermediate eccentricities, the flow transitions to slug flow first.

Figure 11 .
Figure 11.Comparison of the variance of the two differential pressure signals at different eccentricities.

Figure 12 .
Figure 12.Confusion matrix of flow regime clustering results and visual recognition results.

Table 1 .
Summary of gas-liquid two-phase flow research in vertical annular spaces.